Duality and Geometric Programming

DUALITY AND GEOMETRIC PROGRAMMING

G. Reklaitis, V. Salas, A. Whinston

November 1975 WP-75-145

Working Papers are not intended for dis- tribution outside of IIASA, and are solely for discussion and information purposes.

The views expressed are those of the

authors, and do not necessarily reflect

those of IIASA.

Duality and Geometric Programming

G. Reklaitis

Department of Chemical Engineering Purdue University

v.

^Salas

Krannert Graduate School of Industrial Administration Purdue University

A. Whinston

Krannert Graduate School of Industrial Administration Purdue University

October

1975

i

1. - Introduction and Motivation

The theory of production refers bas ically to the problem of optimal allocation of resources or factors of production' such that the total cost of producing a certain output is minimized. If y is an aggregate measure of output that can be produced from a given set of inputs (xl' ••• , x

n) in certain amounts specified by the technical characteristics of the production function y =

F(X

_l , ••. , x

n), and Pl' •.• , P

n are the prices of the inputs, the problem is mathematically formulated by rl, p.

60]

I'

min

X

C =A+'"n_{} . i}px_i i=l

(1.1) subject to

F(X

_l , ••• , x

n) = y (constant)

If the production possibility set allows an output to be produced by an infinity of combinations of productive factors, it would be impossible, without any other considerations, to determine the total cost uniquely for each output. However, the minimization problem in (1.1) eliminates indeter- minacy, so that by soll/ing (1.1), an optimum value for each factor can be obtained as a function of input prices and output.

(1.2)

This gives

C*(y, p) = A +

f

^{p.x.* = A +}

f

p.g.*(y, Pl' ••• , Pn)

i=l 1 1 i=l 1 1

[1, p.

59J

The cost minimization problem gives a total cost function. At the same time, it has pointed out the existence of a dual problem which would allow for determination of production structures from cost curves (2, p.

159].

Extensions of the dual relationship are given by Uzawa

[3]

and Diewert

[4].

The solution of the cost minimization problem is conditioned by the form of the production function representing the underlying technology.

The first form of the production function was thAt of Cobb-Douglas

r5].

In 1961, Arrow et. ale [61 introduced the Constant Elasticity of SUbstitution

(CES) production function. More recently, Christensen et'. ale

[7J, [8]

pro- posed a form for the production function based on a second order Taylor's expansion, evaluated at xi

=

1, of ~ arbitrary explicit production :func- tion. For example, the two inputs one output formulation would be

log Y

=

^{log So + 6}₁ ^{log xl + 6}₂ ^{log x}₂ ^{+ B} 2

3(log xl - log x₂) (1.4) The rationale for the Transcendental Logarithmic Frontier, a.s it is ca.lled, is based on the argument of generality and absence of assumptions that were included in previous representations of production fUnctions. This absence allows the assumptions made in previous forms to be subjected to statistical tests. The hypotheses which have been tested are those derived from the theory of production (hanogeneity, symmetry, and narmalization), and others incl.uded implicitly in tbe Cobb-Douglas and CES farms (additivity and separa- bility of inputs and outputs)

[81.

Two main problems arise from the use of the Transcendental Logarithmic form:

1. - For practical and estimation PJ,rposes, the authors take the approxi- mating :function as the true function and include any possible source of

error in the error term of the regression equation. This implies that

there is no way of telling whether the results

are

affected by stochastic or approximation error.

2. - The Cobb-Douglas and the CES production :function have the property of "self duality", i.e., both the production and the cost forms are members

of the same family of functional forms. This makes irrelevant the choice of representation of the technology by the production or cost functions.

The Transcendental Logarithmic Form when taken as the true form for the pri-, mal (dual) problem and then taken again as the true form of the dual (priJna.l) , makes one of the selections arbitrary since the form is not self-dual. This point is treated by Burgess r9] who shows with empirical results the conse- quences of choosing the cost or the production Transcendental Logarithmic

form as a representation of the underlying technology.

This paper is addressed to the possible solution of these two problems while still being able to work with more general production functions. We propose for the consideration of the economists interested in the Theory of Production, the Geometric Programming (GP) method of solving cost minimization problems which is extensively used in engineering. The similarities observed in both fields also indicate the possible benefits of closer communication among them. In the coming sections, we give an introduction to GP and illus- trate with examples using the Cobb-Douglas, CES, and a more general explicit production function.

2. - Introduction to Geometric Programming

The field of Geometric PrOGI'amniDt; ca:1 be considered initiated with the work of Duffin, Peterson, and Zener, which is sUDlIIlBrized in their book, "Geo- metric Programming"

(13J.

Another valuable reference is Wilde and Beightler

r14,

especially Chapter

4J,

and more recent discussions can be fOWld in

(15].

As is pointed out in

[13,

Chapter

1):

GP "has developed with problems of engineering design ••• (as) an attempt to develop a rapid systematic

method of designing for minimum costs ••• The basis of the method is a relent- less exploitation of the properties of inequalities."

•

The method of GP is particularly suitable for cost functions having polynomial form, with each term of the polynomial being the joint product of a set of variables raised to arbitrary powers. For example, in engi- neering design, the total cost c is a sum of component cost u

i ; i.e.,

c = f ^u. (2.1)

. 1 ~

~= -

where each u

i is a positive function of the design variables tl' ••• , tn'

ot

the form

a ..

u; = c.

H ^t.

~J ... ~j=l J

The c. and a .. are specified parameters. Generalized polynomial inequality

~ ~J

constraints of either sense can also be handled [14, Chapter

Jr.].

The problem of minimizing a polynomial c subject to polynomial constraints is termed a primal program. If a solution to the primal problem exists, there exists a related maximization problem which is called a dual program.

The relation between the primal and dual programs is precisely the resuJ.t of the geometric inequality [13, pp.

4

and

51

m

m

6_i

r:

^5.

u.

~

n

U.

i=l _{~ ~} i=l _~

where U. are arbitrary non negative numbers and 6. are positive weights

~ ~

satisf'ying

If

^{6. =1}

i=l ~ If we let u

i = 5 iU

i , then (2.3) converts to

( ^~~i.)

~

i

~

^u.

~ ¥t

^u

i=l ~ i=l

(2.4)

(2.6)

and if u. is of the form given in (2.2), the right side of (2.5) can be

~

written as

m (c.)

~ⁱ

R

n 2. ^t

i=l 5i j=l j

If the 5 i are chosen so that

f

^{fl .}a.. = 0, for all j . then the function i=l ~ ~J

obta.ined is independent of t. and is called the dual function, V(6):

- c. 6_i J

v(~) = W (2.)

^(2.7)

i=l 5_i

For any set of 6_i satisfying the nornality (

.L

m

r

^1\i _ ,^-

1, ...""

O~~ +~_ ~~~_"uc VJ.' IIUU-

i=l

gonality (

f

^{5 .a .. ==} 0) conditions, the value of v(5) is a lower bound of

i=l ~ ~J

.

^''

the total cost c, and for the 5

i values resulting from maximizing (2.7) sub- ject to normality and orthogonality conditions, the values of the primal and dual objectives are the same (see [131, [14] for the proofs).

It is of interest for the subsequent development to summarize the dual GP problem of a primal minimization problem with constraints:

Suppose a cost function gO(t) is to be minimized subject to a set of constraints _{~(t) ~} 1, k = 1, ••• , p, t 1

j

>

0 where the _~(t) are of the form

~

(t) = r.

c.

R ^t.

aij

-K iErkl _~ j=l J • If _~ (k = 0, ••• , p) is the number of terms in :function k and if all c. are positive

~

as [13, p.

78]

p a.ndm=

r

~

k=O

6., k = 1,

~ ••• , p

(2.8)

subject to _{~ ~i}

=

¹

iEr₀₁

f

^{a ..Fi}_i

⁼

^{0, j = 1, ••• , n}

i=l l.J

The relationship between primal and dual variables at their respective opti- mum values is given by [13, p. 81'

D t

&.:

c. n . ^l.J

l. _J=. 1 J =

fJ •_l.

*

i

e ^[OJ

i

e [k'

mathematical artifice but has engineering interpretations.

where 5* means evaluated at optimum.

Note that the logarithm of v(5) is a concave functicn. Hence, the GP duality theory allows the use of a linearly constraint concave dual maximi- zation problem to solve the nonlinear nonconvex primal. Therein resides the real power of the method. The effort required to solve the dual is related to a parameter called degree of difficulty of such a program, which is given by the number m - n - 1, where m is the total number of terms and n the rank of the exponent matrix. This degree of difficulty is in fact the difference between the number of variabJ.e.s and constraints in the dual program.

When the degree of difficulty is zero, the solution is directly obtained by solving the system of constraints of the dual program. For higher degrees of difficulty, the optimal solution is not as straight-forward, but formalized procedures have been developed to either approximate upper and lower bounds to the cost function [13, pp. 81, 101] or else to iteratively search for the maximum.

As pointed out in [13, p. 13J, the dual problem is not just a The weights 6.

l.

have a one to one corre spondence with the polynomial terms of the prima.1 prob-

lem, and the optimal F..

*

provides the relative size of these terms. The dual

1

problem also has intrinsic features which supply qualitative information about the primalo

We hope to confirm this in the next sections when we use GP to derive some results of the Economic Theory of Production.

3. -

Application of GP to the problems of the Economic Theory of Production

3.1 Illustration with the CObb-Douglas Production Function A Cobb-Douglas Production Function is of the form

n

y

= ^F(X

_l , ••• , x )

= n

^x.ai

n . 1 1

1=

where y is considered an aggregate measure of output, x. is the value of the

1

input i, and ai are Parameters satisfying the condition

f

a. = 1, in order i=l ¹

for the f'uncti on to be homogeneous of degree one.

If it is assumed that the behavior of a firm is directed to minimize the input cost to produce a certain level of output, y, the firm's cost minimiza- t10n problem can be formulated as (primal problem)

n

min I: p.x.

. 1 1 1 1=

n

subject to:

n

^x.ai:it

y,

^{x. :it 0}

i=l 1 1

If we transform the constraint to its equivalent form:

n

y IT

x. -a

i ~ 1 i=l 1

then we can construct the GP dual:

61 6 +1

n

(Pi) (y)

^{n /)}

max v(6) =

n '6

^{0 /) +1 n+l}

i=l i n+l n

subject to:

f

^{5. = 1}

i=l _~

i

=

1, ••• , n 5. :it 0

~

Summing over constraints

(3.1.5),

we have:

f (,

^{i -}

_~

^{n+1 (}

_~ ex.)

= 0

i=l _ i=l _~

which implies that 5 +1 = 1 and (,. * =_n

ex.

~ ~

and by the property of equality between pri.ma1 and dual objectives at opti- mality,

E

^p.x.

i=l ~ ~

where c(p) =

ex.

n ~Pi) ~

=

y n -

i=l i

n

p.

ex

_i

.TI

(;~)

would be the unit cost.

~=l ~

From the correspondence between primal and dual variables, we see that p.x.~ ~

6i* =

n

=

ex.,

~ and also 5. * =~

ex.

~ =

l,; p.x.

i=l _{~ ~}

(\ log Y

o

^log

^x.

_~

or the optimal cost share is equal to the output elasticity with respect to the input 1.

3.2 Illustration with the

CES

Production Function

The primal cost minimization problem for this case will be n

min

^I:

P.x.

x i=l _{~ ~}

) [ n

-b

-lIb

subject to F(x = L a.x

i '] :it Y i=l _~

where I:n a

=

^1.

i=l i

Formulated as a GP primal, the problem becomes n

min

E

p.x.

x

i=l _{~ ~} subject to

b(

ⁿ

b)

y

r: a.x.

i=l ~ ~ ^~ 1

And the corresponding GP dual is 6i b 6n+_i

"":'

i~l (:~) ~ C~:) (i~l

one,

n

subject to

r:

^{5. = 1}

i=l _~ 6_i - b6

n-ti

=

^{0 i}

=

^{1, ••• , n}

summing over i in the second constraint and making use

n 1

we have ~ (, +. = -b • . 1 n ~

~=

of the first

The problem simplifies to

6_i 6_{i /}b

n (Pi) n (a

^{i )}

maxy

n - n -

6

i=l

6

_i i=l

6

_i

n

subject to

r: (,.

= 1 • i=l ~

n

:; max

Y TI

5 i=l

lib 6 1 (

p.a_i ) 6

~

^(l+b

^)/b

~

The solution to problem (3.2.4), obtained via the generalized arithmetic/

geometric mean inequality as shown in Appendix 1, is,

6

*

=

i ~ ^{b/l+b l/l+b}

' L , p. a

i=l _~ i

For this optimal value, as shown in Appendix 1, the minimum of the primal problem is given by

c(y,p)

=

y (

~

(p.a.l/b)b/l+b)l+b/b i=l _{~ ~}

and hence

n 1 _ ( n lib b/l+b)l+b/b

~ p.X. - _~ (p.a. )

. 1 1 1 . 1 1 1

1= 1=

n 1 1

where .1: Pixi is now the normalized unit cost (Xi = Xi/Y).

1=1

Note that the results above can be generalized to any homogeneous pro- duction function of the form

F( )x ^{= [ n}~ a,x ^-b/V]-W/b . 1 1 i

1=

where v and Ware positive parameters.

3.3.

More general results on primal dual relationships

In the previous sections, we have illustrated the use of GP in solving cost minirn.1zati()n .l?roblcms under the differen.t production technologies long used in the study of the theory of production. The effectiveness of the method is particularly clear in the Cobb-Douglas form. In that case, the dual problem has zero degrees of difficulty which allows the dual cost func- tion to be obtained merely :£'rom the solution of the constraints of the inter- mediate GP dual. The GP formulation also illustrates that the optimal cost

shares are independent of the input prices and pro-portional to the elasticity of output to input, O'i. The price independence is generalizable to all the cases of zero degrees of difficulty as is also shown in

(16).

For the CES form of the production function, the dual. problem does n~

have zero degrees of difficulty, but we can still solve for the optimal 6 i

*

by making use of one of the commonly used GP relationships. The optimal 5 .

*

may also be considered a form of writing the demand equation for factor

1

i which in this case is dependent on the inputs prices.

The previous results also suggest more general relationships between the primal and dual problems and fUrther extensions of the role of the inter- mediate GP dual in solving for the input demand ~quations.

If we write the Kuhn-Tucker necessary conditions far optimality for the dual GP problem as stated in (2.8) but with v( 5) replaced by log v( a ), we obtain

o

^log v(.~)_ Anq(~) ~ 0

00 0&

( 0

log _v(~) _ _{~ aq(~)\5}

=

⁰

00 00 ")

q(~)

=

⁰

o

^{:t 0}

where q(6) represents the set of normality and orthogonality constraints;

and A is the associated vector of multipliers. Now, since log v(~) is a concave function, the problem of maximizing log v(o) subject to the linear dual constraints is a concave program. Consequently, the Kuhn-Tucker condi- tions are also sufficient for optimality, and the solution of equations

(303.1)

will be a global maximizing point. In fact, providing that the dual constraints are linearly independent, it will be the unique global maximizing point ••• Next, since it is easily shown that v(a) and log v(a) have the same set of maximizing points,

[13,

Theorem

3.2J,

it follows that the solu- tion to equations

(303.1)

will be the global maximizing point of the dual GP problem (2.8)⁰ Finally, from the duality theory of GP, such a solution will exist providing that the primal constraints possess an interior point and that a feasible minimiZing solution to the primal exists. Moreover, at

their respective optimia, the primal cost dual objective function values will be the same and the respective variables will be related via equations (2.9).

In the case of our cost minimization problem, the objective function is always linear and positive, and the problem always involves only a single posynomial constraint. Hence, an interior point .can always be found, and a minimizing solution will exist providing the problem is bounded. Hence, under reasonable conditions, a solution to _e~uations

(3.3.1)

can always be found. In general, that solution, 5*, will be a function of p, although only in special situations will it be possible to solve

(3.3.1)

to obtain

,

an e..'q>licit functional form b* = f(p). If such a functional farm can be determined, then when 1)* is substituted into the GP dual objective function, the dual cost function in the Shephard sense, C(y,p) will be obtained. From the GP duality theory, we have assurance that this dual objective function value will lJe exactly e~t1.al to the priLlal objective value evaluated at its rrdnim.lzint; point, ;.e~;

~n p.x.

=

v(S*)

=

^{C(y,p) •}

i=l ].].

Taking the derivatives with respect to p in

3.3.2,

we have oV(5*) d6

=

~C(y,p)

=

x

06* • dP dP •

Dividing both sides by V(5*) and nmltiplying by p, we obtain,2

(0 log v(5*)

M.) _

^*

P 08* •

op - °

⁰

where 5~ is the subvector consisting of the first n components of 6*. From the e~uivalencebetween the primal and dual solutions, _6~ will be the same as the first n canponents of the Ii* obtained by so1.ving (3.3.1.).3

As an illustration, we can take the CES case. Equation

(3.3.2)

for that case is written as

n

Y(

^{n 1./b b /1.}

+b)~

~ p:x

i

=

E ( p . a . ) b

i=1.]. i=1.]. ]. •

','

Taking derivatives with respect to p., we have

J.

n

I /

^{l+b / b 1}

_ ( ( 1 b)b l + b ) - - 1 ( 1 b ) - - 1 -

x. - ! p.a. b p.a. l+b a. b

J. ' 1 J.J. J . J . , J.

J.=

or

=

1/b-E-

(p.a. )l+b

J. J.

n lib

....E..-

I: (p.a. )l+b i=l J. J.

which is the same as (3.2.5).

The results above show how the intermediate GP dual can provide the equivalent demand equations for factor i without having to actual.1¥ write the explicit cost function and then take the derivative with respect to p.

They can easily be extended to the case when (3.3.1) does not have a solu- tion for {) in 1#erms of p only, because of non 1inearities in (3.3.1) which do not permit the elirunation of A. In this case, A will appear in C(y,p,).), and the composite function may even be difficult to write explicitly. But since the results (3.3.2) to (3.3.1.1-) still apply, and if we are interested in the form of the demand equation, as most empirical studies are

r ⁸ l, r91,

then C(y,p,).) does not have to be computed since we show that the same result is obtainable by simply using the intermediate GP dual. Note that the

results are independent of the condition of self-duality of Production and Cost :t'unctiona1 forms which in fact, restricts attention to only a particu- lar class of functions.

3.3.1 Illustration with a General Production Function Consider the concave prod~ctionfunction

y

=

^F(x)

=[ _{~ ~}

^c.

x.-~/2 x.-~/2J-l/~

i=l j=i J.j J. J

where y represents again aggregate output and x. are the input factors

~

(i

=

^{1, ••• ,} n). The cij and ⁰¹ are parameters, and the function is homogene- ous of degree one.

The reasons for selecting the above form are:

1 - It has input interaction terms that will allow for testing sane

assumptions implicit in other production functions (like separability on inputs of the Cobb-Douglas and CES).

2 - It has the property of approaching in the limit a Cobb-Douglas form when ^{01 -.}

o.

From

(3.3.1),

the elasticity of output to input

x.

would be

~

Ji f

c

-01/2 -01/2)

a

log ^y = Xi

oY

= 2\j=i ijxi xj

n ~

-01/

²

-01/

²

~ log Xi Y OXi i~l j~ cijxi x j

The cost minimization problem under

(3.3.1.1)

would actual.l¥ be min

x

n

~ PiX.

i=l ~

subject to [

~ ~

^{cijx.-01/2 x.}

^-ot/2J-1/

^0I

^a

^y

i=l j=i ~ J

x.

_~ a

0

i

= ^{1, ••• , n}

where Pi are the input prices (Pi> 0 for all i). The competitive equili- brium conditions would be, using

(3.3.1.2)

f

^-01/2 -01/2

p.x._{~ ~ _ =1}j .c ..x._{~J ~} x._J ---- - n

-01/2 -01/2

Pk~ !: c._.:~

x.

j=k A.J J

Equation

(3.3.1.4)

couJ.d be used in estimating the parameters c

ij and tJI

and in testing certain assumptions on them. However, this would require the use of non-linear estimat:l.on procedures.

.~

The GP dual of (3.3.1.3) woul.d be written as

6 ex 6

(~ f

^{& )}

n

(PO)

^{i n ( n ( Co}

⁰⁾

^{ij) ( n n )}

⁰

^{-1. j-1 ij}

max IT.2:. IT 11

y

^{2:.J.. L}

r

^{6 J.- - .}

5 i=l ~i ^{i=l j=i} 6ij i=l j~l ^ij

n

subject to

r

^{&i = 1}

i=l

~ 0

a

0, 60j :a 0

J. J.

i

=

^{1, 000, n}

i,j

=

1, ••• , n and j

a

i where J(i) is a subset of the set of subscripts pairs (h,.t) with

t

a h,

such that either h = i or t

=

i but not both.

More explicitly, for the three input case of the form

(3.3.1.5) woul.d read

3

Pi 6i

3 3 I'c

_ij ⁶ⁱ

;l, 3 3 (i~l

~ i~l ⁽⁵⁾ i~l (j~

⁽

~ij)

⁾

(i~1 j~i

^{6ij )}

3

subject to L 50 = 1 i=l J.

5 i a 0, &ij

a

0

(3.301.8)

summing over constraints (303.1.9) to (3.3.1.11) and using (3.3.1

.8),

we have the result

3 3 3

I: 0i -

0'(

^J.

^r ~..) ⁼ °

i=l i=l j=i J.J

3 3

and I: I: _~tj i=l j=1 J.

= -1

Considering the equivalence between primal and dual variables:

3

Pix. = _6~

^I:

P.x.)

i

= 1, 2, 3

J. {\i=l J. J.

-0'/2 -0'/2 _ _~tj

cijxi xj - -3",.----:;;;~3--

L

r.

i=l j=i

i

=

1, 2,

3

i, j = 1, 2,

3 ,

j :l i

(3.3.1.16)

we can show that constraints

(3.3.1.9)

to

(3.3.1.11)

are in fact the com- petitive equilibrium conditions as expressed by

(3.301.4).

Note fron

the

e<;i.uiv~lt:;nce relations,

(3.3.1.15 - .17),

that xi

>

0,

i = 1, 2,

3

if and only if 0i' 0ij

>

0, i,j = 1, 2,

3,

^{j :l} i. Consequent~, for xi

>

0, the Kuhn-Tucker conditions

(3.3.1)

collapse to the conventional

Lagrangian necessary conditions.

For

the case of problem

(3.3.1.7),

these

~e,

log c'_J.i - log A.. -_J.J. ^Q'

A2 - 1

i

=

^{0 i}

= 1,2,3

log C. - log

~

^{.. - 9:}

(L l~) - ¹ =

^{0 i,j}

= ^{1,.2, 3}

J.j J.J 2

hEH(i,j i"

j :l i together with equations,

(303.1.21)

where Xl and A2 are the La.grange multipliersi and where H(1,j) is the set of all h such that (i,j) is in J(h).

If we solve

(3.3.1.18)

to

(3.3.1.21)

in terms of

. 0.,

^~ i =

1,2, 3

and ucc the results hi cq ...l..1.tions

(].3.1.:;)

to

(3.3.1.11),

\ve obtain tt.e foUol'l- ing system of equatioas in _~i' i

=

1, 2,

3

and Pi' i

=

1, 2,

3.

(See Appen- dix 3 for the derivation.)

(p

^O22)

^(& 3)

Bll ^{+ ~12Pl +} 8

13

^{+ 8}

14

^P

3 -

0

f3 21

⁺

62~1

⁺

^{6 23} ~1)

₁ ⁺

^{624 (:3)} ₃ ⁼

⁰

S31

+

~32P3

⁺

633 (;1)

₁ ⁺

~34 ^(;2)

₂ ^{= 0}

with the restrictions 8

12 = f3 22

⁼

a ₃₂

613 - f3 23 =

0

1=31

4 - 8 33 =

0

~24

- f3 34 =

0 where

1

= _~,

2, 3

/6

_i

4 =

-OleA1

6 =

c

e-«(:I/2

ij ij

i = ~, 2,

3

1)( _~ A2 )h + Xl + _~ i _~ , 2

3 __

^A _~ 1

hEH(1,j) ,~

=., ,

^{~ ~}

^>

The systen

of

€~~tions

(3.3. 1

.22)

allows

us to solve

tar

8

1, 1

=.1, 2, 3

in terms of Pi' i

=

1, 2,

3,

Al and

A~'

ⁱ

=

1, 2,

3.

The final system is, however, non-linear in Al and

A~'

and these variables can not be elimina- ted in such a wa~- that (). becomes a function of p. alone. If we a.ssume

~ ~

the economy is operating at optimum, by takinr; data on 5., cost share,

~

and Pi' factor prices, and treating Al and

A~

^as

par~neters,

we can use sys- tem (3.3.1.22) with the restrictions (3.3.1.23) to estimate B.. by statisti-

J.J

cal procedures using other functional forms _[8J~

[9].

Likewise, statistical tests on assumptions about production technology could be performed. For example, to test the assumption on input separability, we would test for

4. -

Conclusion

In this paper, we have studied the problems of finding the dual cost function associated with a particular production technology and the deriva- tion of the demand function of a factor i with the methodology of GP.

We used some results from GP to illustrate the prima.l dual relationships and to show how the intermediate GP dual can replace the' so-called

Shephardfs dual cost function C(y,p) for empirical. studies concerned with the demand equation for factor i. An explicit production function with interaction terms among the factors has been used to illustrate same of the results and to show how an explicit general form can be used to test same of the assumptions of' the theory that before had been tested with appraxi- mated and, in some way, arbitrary forms.

It is also important to point out that if we start with the cost tunc- tion explicitly and write the dual problem (2)

(4.1) the GP method that in the paper has been used to solve the primal problem to (4.1), would still be applicable to the solution of the above problem.

As a corollary, the paper supports an idea introduced already in

[171

of the utility of using some of _t~e concepts developed in the engineering field to model the Economic system, since a closer look reveals that both fields are looking at similar problems.

5. -

Footnote s

1. i E [k] indicates the range of values for i in the kth constraint; that

k-l k

is, i going from 1: ~ + 1 till 1: ~, whE;re ~ is the number of

k=O k=O

terms of the kth constraint.

2. The multiplication by p is in the form of Kroeneker product

® ;

that is, we multiply by p., i = 1, ••• , n, the respective ith element of the

1.

vectors at both sides of equation

(3.3.3).

3.

The remaining components of 6 are the dual variables associated with the constraint terms. In the C-D and CES cases, these could be eliminated by means of the dual constraints. In general, they are of course sJ:ways available as part of the optimal dual solution.

6. -

Bibliography

1. Samuelson, P. A.. Foundations of Economic Analysis, Atheneiun, New York, 1967.

2. Shephard, R. V., Theory of Cost and Production Functions, Princeton University Press,

1970.

3.

Uzawa. H., "Duality Principles in the Theory of Cost and Production,"

International Economic Review May 1964, Vol. 5. No.2, pp. 216-220.

4. Diewert, \'1. E., An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function," Journal of Political Economy 79, J.1ay/June 1971.

5. Cobb, C. W. and Douglas, P. H., "A Theory of Production," American Economic Review. Vol. 17. No.1, Supplement, March 1928.

6. Arrow, K. J., Chenery, H. B., Minhas, B. S., and Solow, R. N.,

"Capital Labor Substitutions and Economic Efficiency," The Review of Economics and Statistics, Vol. 4?, No.3. August 1961.

7. Christensen, L. R., Jorgenson, D. W., and Lau, L. J., "Conjugate Duality and the Transcendental Logorithmic Production Froutier,"

Econometrica Vol. 39, No.4, July 1971.

8.

Christensen, L. R., Jorgenson, D. W., and Lau, L. J.,

"Transcendental Logorithmic Production Frontiers," The Review of Economics and Statistics. Vol. 55, No.1. February

1973.

9. Burgess, D. P., "Duality Theory and Pitfalls in the Specification of Technologies, " Journal of Econometrica, Vol. ~, No.2, May 1975.

10. Christensen, L. R. Jorgenson, D. W., and Lau, L. J., "Transcendental Logorithmic Utility Functions," Institute for Mathematical Studies in the Social Sciences, Stanford University, Technical Report No.

94,

March 197:'.

11. H::mthakker, H. S., "A note on Self-Dual Preferences," Econometrica, Vol.

33,

No.4, October 1965.

12. Samuelson. P. A., "Using Full Duality to Show that Simultaneous'by Additive Direct and Indirect Utilities implies Unitary Price Elasti- city of Demand, Econometrica, Vol.

33,

No.4, October 1975.

13. Duffin, R. J. Peterson, E. L., Zener, C, Geometric Programming, John Wiley and Sons, Inc., New York, 1967.

14. Wilde. D. J., Beightler, C. S., Foundations of Optimization, Prentice Hall, Inc. New York, 1967.

15. Avrie1, M., Rijckaert, M. J., and Wilde, D. Jo, OPtimization and Design, P".centicE:: Hall, Inc., 1973.

16. Theil, H•. "Substitutbn Effects i"-1 Ge:JmetriC' Pr()gramming." Management Sdenre. Vol. 19 no. 1. September 1972.

17. Marsden,

.T.,

Pingry, D., and Whinston, A., "Engineering Foundations of Production Functions," Journal of Economic Theory, Vol. 9, No.2, October 1974.

7. -

Appendix 1

Solution of the problem (3.2.4) in section 3.2 using the Generalized ArithmeticJgeametric mean inequal'ity

Given xi

>

0 and

~i

^{:It 0, i}

=

^{1, ••• , n,}

~

^O'i

=

^{1, for any r}

>

0

(Al.l)

(Al.2)

•

with equality if and only if all x. are equal. For a.rry s

>

0, defin-

J.

ing Y

i

=

^{XiU'iS' i}

=

1, ••• , n (Al.I) convert.s to ()i

[~

Q'il-rs yirj/r :l

~ (Y

ⁱ

S)

J. J. ~i

Since I: ct. = 1, (Alo2) is satisfied as equality if and only if for

• J.

J.

i

=

^{1, ••• ,}

^n,

Y_il/s

~i = ---=-l/""s-- 'E y.

i J.

for any s. (Al.3)

SUbstituting this value of

ex.

into the left side of (Al.2),

J.

-l-rs r (I: y. s Y.)

i J. J.

( l/s)l-rs

~

Y

_i

J.

l/r

=

(Al.4)

l+b l/b

In problem (3.2.4), s

= b

and Y

i

=

^Pia_i ,which when sub- stituted into (AI. 3) and (Al.4), just!fies the re sul.t s shown in the main text, respectively, (3.2.5) and (3.2.6).

Appendix 2

Limit results for function

(3.3.1)

We have the function:

y = F(x) =

[~ ~

^{c ..x.}

-ex/

^{2 x}

-a/

²

J-

^l

^/0/

i=l j=i ~J ~ j

(A2.1)

c..x. -ex/2J

~J ~

n l:

j=i

ex

Taking logaritmns , n log [ L

log Y = -

- - - -

i=l

^... _----

°

If a ~ 0, lim (log y) =

0'

^if

a-+O

n n

l: l: cij

=

^1.

i=l j=i

For re solving the indeterminacy, we use 1 'Repital's rule

n n

lin (log y) 5 lin

(_%a

^(log

[.t ^.t

~=l J=i

ctJA) ~ ---::rOcx~7~0ct---

Taking the derivatives:

~ ^{~( ~ ~ ~}

^..x.

^-ex/

² x. -0//2 log (x .• x.»)

i=l '=i ~J ~ J ~ J

log Y = lim ---:;....;;;....:J~. _

_.•n n n

/2 /2

~ ~ ~

c .x.-O/

_x.~

i=l j=i iJ ~ ^J

(A2.4)

which is equal to

n n

log y = _~

t

c

ij log (xi· x J.) i=l j=i

(A2.5)

which can always be written as a Cobb-Douglas by choosing c

ij such that

n n

~

r

^c_iJ^{' = 1.}

i=l j=i

Appendix 3

Solving 0i in terms of

A1 , A~'

^{and Pi in}

(3.3. 1 .9)

to

(3.3.1.11)

i

=

1, 2,

3

i

=

^{1, 2,}

³

which implies that

i

=

1, 2,

3

Also, from

(3.3.1.19)

and

(3.3.1.20), .

and

i,j = 1, 2,

3

j ;a i

(A3.4)

or,

^using

(A3.2) above,

i,j = 1, 2,

3 •

j

a

i

With these results, the system

(3.3.1.9)

to

(3.3.1.11)

becomes,

(A3.6)

whichcarresponds to (3.3.1.22) with (3.3.1.23) and (30301.24)

ⁱⁿ

the main text after dividing each equation in A306

^by

5 ii

^ote

Alo